We will now look at what is known as The Sequential Criterion for Limits at Infinity and Negative Infinity, which is analogous to the other sequential criterion theorems.

Theorem 1: Let $f : A \to \mathbb{R}$ be a function and suppose that $(M, \infty) \subseteq A )$ for some $M \in \mathbb{R}$. Then $\lim_{x \to \infty} f(x) = L$ if and only if for every sequence $(a_n)$ from $A$ such that $\lim_{n \to \infty} a_n = \infty$ we have $\lim_{n \to \infty} f(a_n) = L$.

$\Leftarrow$ Suppose that for all sequences $(a_n)$ from $A$ such that $\lim_{n \to \infty} a_n = \infty$ we have that $\lim_{n \to \infty} f(a_n) = L$. We want to show that $\lim_{x \to \infty} f(x) = L$.